MATERIALS FOR HEAT EXCHANGER

Heat exchangers take heat from one fluid and pass it to a second (Figure 1). The fire-tube array of a steam engine is a heat exchanger, taking heat from the hot combustion gases of the firebox and transmitting it to the water in the boiler. The network of finned tubes in an air conditioner is a heat exchanger, taking heat from the air of the room and dumping it into the working fluid of the conditioner. A key element in all heat exchangers is the tube wall or membrane that separates the two fluids. It is required to transmit heat, and there is frequently a pressure difference across it, which can be large.

Figure 1 A heat exchanger. There is a pressure difference Δp and a temperature difference ΔT across the tube wall that also must resist attack by chloride ions.

Table 1 Design requirements for a heat exchanger

Function	Heat exchanger
Constraints	- Support pressure difference, Δp - Withstand chloride ions - Operating temperature up to 150o C - Modest cost
Objective	- Maximize heat flow per unit area (minimum volume exchanger) or - Maximize heat flow per unit mass (minimum mass exchanger)
Free variables	- Tube-wall thickness, t - Choice of material

The model

First, a little background on heat flow. Heat transfer from one fluid, through a membrane to a second fluid, involves convective transfer from fluid 1 into the tube wall, conduction through the wall, and convection again to transfer it into fluid 2. The heat flux into the tube wall by convection (W/m²) is described by the heat transfer equation:

q = h_1­ΔT₁

 in which h1 is the heat transfer coefficient and ΔT₁ is the temperature drop across the surface from fluid 1 into the wall. Conduction is described by the conduction (or Fourier) equation, which, for one-dimensional heat-flow takes the form:

q = λ (ΔT/t)

where λ is the thermal conductivity of the wall (thickness t) and ΔT is the temperature difference across it. It is helpful to think of the thermal resistance at surface 1 as 1/h1; that of surface 2 is 1/h2; and that of the wall itself is t/λ. Then continuity of heat flux requires that the total resistance 1/U is 

where U is called the ‘‘total heat transfer coefficient’’. The heat flux from fluid 1 to fluid 2 is then given by

q = U (T₁−T₂ )

where (T₁−T₂) is the difference in temperature between the two working fluids. When one of the fluids is a gas—as in an air conditioner—convective heat transfer at the tube surfaces contributes most of the resistance; then fins are used to increase the surface area across which heat can be transferred. But when both working fluids are liquid, convective heat transfer is rapid and conduction through the wall dominates the thermal resistance; 1/h1 and 1/h2 are negligible compared with t/λ. In this case, simple tube or plate elements are used, making their wall as thin as possible to minimize t/λ. We will consider the second case: conduction-limited heat transfer, where the heat flow . Consider, then, a heat exchanger with n tubes of length L, each of radius r and wall thickness t. Our aim is to select a material to maximize the total heat flow: 

where A=2πrLn is the total surface are of tubing. This is the objective function. The constraint is that the wall thickness must be sufficient to support the pressure Δp between the inside and outside, as in Figure 1. This requires that the stress in the wall remain below the elastic limit, σy, of the material of which the tube is made (multiplied by a safety factor—which we can leave out):

This constrains the minimum value of t. 

The heat flow per unit area of tube wall, Q/A, is maximized by maximizing

M₁ = λσ_y

Four further considerations enter the selection. It is essential to choose a material that can withstand corrosion in the working fluids, which we take to be water containing chloride ions (sea water). Cost, too, will be of concern. The maximum operating temperature must be adequate and the materials must have sufficient ductility to be drawn to tube or rolled to sheet. Cost, too, will be of concern. 

The selection 

A preliminary search (not shown) for materials with large values of M1, using the CES Level 1/2 database , suggests copper alloys as one possibility. We therefore turn to the Level 3 database for more help. The first selection stage applies limits of 150oC on maximum service temperature, 30 percent on elongation, a material cost of less than $4/kg and requires a rating of ‘‘very good’’ resistance to sea water. The second stage (Figure 2) is a chart of σy versus λ enabling M1 = λσy to be maximized. The materials with large M1 are listed in Table 2.

Figure 2. A chart of yield strength (elastic limit) σy against thermal conductivity, λ, showing the index M1, using the Level 3 CES database.

 Table 2 Materials for heat exchangers

Material	Comment
Brasses	Liable to dezincification
Phosphor bronzes	Cheap, but not as corrosion resistant as aluminum-bronzes
Aluminum-bronzes, wrought	An economical and practical choice
Nickel-iron-aluminum-bronzes	More corrosion resistant, but more expensive

It is sometimes important to minimize the weight of heat exchangers. Repeating the calculation to seek materials the maximum value of Q/m (where m is the mass of the tubes) gives, instead of M1, the index

M₂ = (λσ_y²)/ρ

where ρ is the density of the material of which the tubes are made. (The strength σy is now raised to the power of 2 because the weight depends on wall thickness as well as density, and wall thickness varies as 1/σy. Similarly, the cheapest heat exchangers are those made of the material with the greatest value of 

M₃ = (λσ_y²)/(C_m ρ)

where Cm is the cost per kg of the material. In both cases aluminum alloys score highly because they are both light and cheap. The selections are not shown but can readily be explored using the CES system.